The
new spectral paradigm of geometry is encoded by Spectral triples \(({\mathcal {A}},{\mathcal {H}},D)\)  where the unbounded operator \(D\)  encodes the analogue of the Dirac operator as described in Section  and where \({\mathcal {A}}\)  is an algebra of operators in the Hilbert space \({\mathcal {H}}\) . The algebra \({\mathcal {A}}\)  is not assumed to be commutative and its inner automorphisms (as explained in Section ) will correspond to the internal symmetries in physics. Because of its flexibility, this new paradigm provides the needed tool to
refine our understanding of the structure of physical space in the small and to “seek the foundation of its metric relations outside it, in binding
forces which act upon it". The main idea, described in details in [1]}, is that the line element now embodies not only the force of gravity but all the known forces, electroweak and strong, appear from the spectral action and the inner fluctuations of the metric. This provides a completely new perspective on the geometric interpretation of the detailed structure of the Standard model and of the Brout-Englert-Higgs mechanism. One gets the following simple mental picture for the appearance of the scalar field: imagine that the space under consideration is two sided like a sheet \(S\)  of paper in two dimensions. Then when differentiating a function on such a space one may restrict the function to either side \(S_\pm \)  of the sheet and thus obtain two spin one fields. But one may also take the finite difference \(f(s_+)-f(s_-)\)  of the function at the related points of the two sides. The corresponding field is clearly insensitive to local rotations and is a scalar spin zero field. This, in a nutshell, is how the Brout-Englert-Higgs field appears geometrically once one accepts that there is a “fine structure" which is revealed by the detailed structure of the standard model of matter and forces. Eventually, this allows one to uncover the
geometric meaning of the Lagrangian of gravity coupled to the standard model. This extremely complicated Lagrangian is obtained from the spectral action developed in [2]} which is the only natural additive spectral invariant of a noncommutative geometry.
